On the separation of solutions to fractional differential equations of order $\alpha \in (1,2)$

Abstract

Given the Caputo-type fractional differential equation $D^\alpha y(t) = f(t, y(t))$ with $\alpha \in (1, 2)$, we consider two distinct solutions $y_1, y_2 \in C[0,T]$ to this equation subject to different sets of initial conditions. In this framework, we discuss nontrivial upper and lower bounds for the difference $|y_1(t) - y_2(t)|$ for $t \in [0,T]$. The main emphasis is on describing how such bounds are related to the differences of the associated initial values.